Quantitative functional calculus in Sobolev spaces
In the frame work of Sobolev (Bessel potential) spaces Hn(Rd,R or C), we consider the nonlinear Nemytskij operator sending a function x∈Rd↦f(x) into a composite function x∈Rd↦G(f(x),x). Assuming sufficient smoothness for G, we give a “tame” bound on the Hn norm of this composite function in terms of...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2004-01-01
|
| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2004/832750 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832560077501890560 |
|---|---|
| author | Carlo Morosi Livio Pizzocchero |
| author_facet | Carlo Morosi Livio Pizzocchero |
| author_sort | Carlo Morosi |
| collection | DOAJ |
| description | In the frame work of Sobolev (Bessel potential) spaces Hn(Rd,R or C), we consider the nonlinear Nemytskij operator sending a function x∈Rd↦f(x) into a composite function x∈Rd↦G(f(x),x). Assuming sufficient smoothness for G, we give a “tame” bound on the Hn norm of this composite function in terms of a linear function of the Hn norm of f, with a coefficient depending on G and on the Ha norm of f, for all integers n,a,d with a>d/2. In comparison with previous results on this subject, our bound is fully explicit, allowing to estimate quantitatively the Hn norm of the function x↦G(f(x),x). When applied to the case G(f(x),x)=f2(x), this bound agrees with a previous result of ours on the pointwise product of functions in Sobolev spaces. |
| format | Article |
| id | doaj-art-f1b15d9528944b2dab64f813b4da529e |
| institution | Kabale University |
| issn | 0972-6802 |
| language | English |
| publishDate | 2004-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-f1b15d9528944b2dab64f813b4da529e2025-02-03T01:28:36ZengWileyJournal of Function Spaces and Applications0972-68022004-01-012327932110.1155/2004/832750Quantitative functional calculus in Sobolev spacesCarlo Morosi0Livio Pizzocchero1Dipartimento di Matematica, Politecnico di Milano, P.za L. da Vinci 32, I-20133 Milano, ItalyDipartimento di Matematica, Università di Milano, Via C. Saldini 50, I-20133 Milano, ItalyIn the frame work of Sobolev (Bessel potential) spaces Hn(Rd,R or C), we consider the nonlinear Nemytskij operator sending a function x∈Rd↦f(x) into a composite function x∈Rd↦G(f(x),x). Assuming sufficient smoothness for G, we give a “tame” bound on the Hn norm of this composite function in terms of a linear function of the Hn norm of f, with a coefficient depending on G and on the Ha norm of f, for all integers n,a,d with a>d/2. In comparison with previous results on this subject, our bound is fully explicit, allowing to estimate quantitatively the Hn norm of the function x↦G(f(x),x). When applied to the case G(f(x),x)=f2(x), this bound agrees with a previous result of ours on the pointwise product of functions in Sobolev spaces.http://dx.doi.org/10.1155/2004/832750 |
| spellingShingle | Carlo Morosi Livio Pizzocchero Quantitative functional calculus in Sobolev spaces Journal of Function Spaces and Applications |
| title | Quantitative functional calculus in Sobolev spaces |
| title_full | Quantitative functional calculus in Sobolev spaces |
| title_fullStr | Quantitative functional calculus in Sobolev spaces |
| title_full_unstemmed | Quantitative functional calculus in Sobolev spaces |
| title_short | Quantitative functional calculus in Sobolev spaces |
| title_sort | quantitative functional calculus in sobolev spaces |
| url | http://dx.doi.org/10.1155/2004/832750 |
| work_keys_str_mv | AT carlomorosi quantitativefunctionalcalculusinsobolevspaces AT liviopizzocchero quantitativefunctionalcalculusinsobolevspaces |